Knowledge organisers

Integrals: CBSE Class 12 Maths knowledge organiser

Everything to know about integrals on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Antiderivative: F is an antiderivative of f when F′ = f; add a constant C.
Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a).
Integration by parts: The rule for integrating a product, choosing u in the order ILATE.

Key formulas

Basic	\(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C\ (n\ne-1),\) \(\int\tfrac1x\,dx=\log\|x\|+C\)
Exp	\(\int e^x\,dx=e^x+C,\) \(\int a^x\,dx=\tfrac{a^x}{\log a}+C\)
Trig	\(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int\sec^2x\,dx=\tan x+C\)
Trig	\(\int\cosec^2x\,dx=-\cot x+C,\) \(\int\sec x\tan x\,dx=\sec x+C,\) \(\int\cosec x\cot x\,dx=-\cosec x+C\)
Trig	\(\int\tan x\,dx=\log\|\sec x\|+C,\) \(\int\cot x\,dx=\log\|\sin x\|+C\)
sec, cosec	\(\int\sec x\,dx=\log\|\sec x+\tan x\|+C,\) \(\int\cosec x\,dx\) \(=\log\|\cosec x-\cot x\|+C\)
Inverse trig	\(\int\frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+C,\) \(\int\frac{dx}{1+x^2}=\tan^{-1}x+C\)
Special	\(\int\frac{dx}{x^2-a^2}=\frac1{2a}\log\left\|\frac{x-a}{x+a}\right\|+C,\) \(\int\frac{dx}{a^2-x^2}=\frac1{2a}\log\left\|\frac{a+x}{a-x}\right\|+C\)
Special	\(\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C,\) \(\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac xa+C\)
Special	\(\int\frac{dx}{\sqrt{x^2\pm a^2}}=\log\left\|x+\sqrt{x^2\pm a^2}\right\|+C\)
Root	\(\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}2\sin^{-1}\frac xa+C\)
Root	\(\int\sqrt{x^2\pm a^2}\,dx=\frac x2\sqrt{x^2\pm a^2}\) \({}\pm\frac{a^2}2\log\left\|x+\sqrt{x^2\pm a^2}\right\|+C\)
By parts (ILATE for the first function \(u\))	\(\int uv\,dx=u\int v\,dx-\int\Big(u'\int v\,dx\Big)dx\)
Special form	\(\int e^x\big[f(x)+f'(x)\big]dx=e^xf(x)+C\)
\(ax^2+bx+c\): complete the square; for \(\frac{px+q}{ax^2+bx+c}\) write \(px+q=A\frac{d}{dx}(ax^2+bx+c)+B\)
Partial fractions	\(\frac{px+q}{(x-a)(x-b)}=\frac A{x-a}+\frac B{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}=\frac A{x-a}+\frac B{(x-a)^2}+\frac C{x-b}\)

More formulas are on the full CBSE Class 12 Maths formula sheet.

Worked example

\(\displaystyle\int\frac{3x + 1}{x^2 + 2x + 5}dx\): \(3x + 1 = \frac32(2x + 2) - 2\), giving \(\frac32\log(x^2 + 2x + 5) - \tan^{-1}\frac{x + 1}{2} + C\).

Common mistakes

Forgetting \(+C\) in indefinite integrals, or keeping old limits after a substitution.
\(\int\frac{dx}{2x + 3} = \frac12\log|2x + 3|\), not \(\log|2x + 3|\).
Applying partial fractions to an improper fraction without dividing first.
Using \(F(b) - F(a)\) when \(f\) is discontinuous in \([a, b]\).

You should be able to…

Integrate the standard forms, use simple substitutions and evaluate basic definite integrals.
Use the special-form integrals, partial fractions and integration by parts on board-standard questions.
Write complete long answers on definite integrals, including the property questions, with each step and the property named.
Handle the hardest property-based integrals and the tricky substitutions that decide 95+.

The printable sheet

Integrals knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Integrals knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers